scholarly journals Characteristics of (γ,3)-critical graphs

2010 ◽  
Vol 4 (1) ◽  
pp. 197-206 ◽  
Author(s):  
D.A. Mojdeh ◽  
P.Y. Firoozi
Keyword(s):  
Edge Connectivity ◽  
Critical Graphs ◽  
Critical Graph

In this note the (?,3)-critical graphs are fairly classified. We show that a (?,k)- critical graph is not necessarily a (?,k?)-critical for k?? k and k,k? E {1,2,3}. The (2,3)-critical graphs are definitely characterized. Also the properties of (?,3)-critical graphs are verified once their edge connectivity are 3.

scholarly journals Split Domination Vertex Critical and Edge Critical Graphs

2020 ◽  
Vol 26 (1) ◽  
pp. 55-63
Author(s):  
Girish V R ◽  
Usha P
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Critical Graphs ◽  
Critical Graph ◽  
Induced Graph ◽  
Vertex Removal ◽  
Edge Addition

A dominating set D of a graph G = (V;E) is a split dominating set ifthe induced graph hV 􀀀 Di is disconnected. The split domination number s(G)is the minimum cardinality of a split domination set. A graph G is called vertexsplit domination critical if s(G􀀀v) s(G) for every vertex v 2 G. A graph G iscalled edge split domination critical if s(G + e) s(G) for every edge e in G. Inthis paper, whether for some standard graphs are split domination vertex critical ornot are investigated and then characterized 2- ns-critical and 3- ns-critical graphswith respect to the diameter of a graph G with vertex removal. Further, it is shownthat there is no existence of s-critical graph for edge addition.

scholarly journals A Better Lower Bound on Average Degree of Online $k$-List-Critical Graphs

10.37236/6405 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Lower Bound ◽  
Average Degree ◽  
Critical Graphs ◽  
Critical Graph

We improve the best known bounds on average degree of online $k$-list-critical graphs for $k \geqslant 6$. Specifically, for $k \geqslant 7$ we show that every non-complete online $k$-list-critical graph has average degree at least $k-1 + \frac{(k-3)^2 (2 k-3)}{k^4-2 k^3-11 k^2+28 k-14}$ and every non-complete online $6$-list-critical graph has average degree at least $5 + \frac{93}{766}$. The same bounds hold for offline $k$-list-critical graphs.

scholarly journals Notes on the binding numbers for (a, b, k)-critical graphs

2007 ◽  
Vol 76 (2) ◽  
pp. 307-314 ◽  
Author(s):  
Sizhong Zhou ◽  
Jiashang Jiang
Keyword(s):  
Critical Graphs ◽  
Binding Number ◽  
Spanning Subgraph ◽  
Critical Graph

Let G be a graph of order n, and let a, b, k be nonnegative integers with 1 ≤ a < b. An [a, b]-factor of graph G is defined as a spanning subgraph F of G such that a ≤ dF(x) ≤ b for each x ϵ V (F). Then a graph G is called an (a, b, k)-critical graph if after deleting any k vertices of G the remaining graph of G has an [a, b]-factor. In this paper, it is proved that G is an (a, b, k)-critical graph if the binding number and Furthermore, it is showed that the result in this paper is best possible in some sense.

MATCHING PROPERTIES IN DOUBLE DOMINATION EDGE CRITICAL GRAPHS

2010 ◽  
Vol 02 (02) ◽  
pp. 151-160 ◽  
Author(s):  
HAICHAO WANG ◽  
LIYING KANG
Keyword(s):  
Perfect Matching ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Critical Graphs ◽  
Critical Graph ◽  
Odd Order ◽  
Even Order ◽  
Double Domination

A vertex subset S of a graph G = (V, E) is a double dominating set for G if |N[v]∩S| ≥ 2 for each vertex v ∈ V, where N[v] = {u |uv ∈ E}∪{v}. The double domination number of G, denoted by γ×2(G), is the cardinality of a smallest double dominating set of G. A graph G is said to be double domination edge critical if γ×2(G + e) < γ×2(G) for any edge e ∉ E. A double domination edge critical graph G with γ×2(G) = k is called k - γ×2(G)-critical. In this paper, we first show that G has a perfect matching if G is a connected 3 - γ×2(G)-critical graph of even order. Secondly, we show that G is factor-critical if G is a connected 3 - γ×2(G)-critical graph with odd order and minimum degree at least 2. Finally, we show that G is factor-critical if G is a connected K1,4-free 4 - γ×2(G)-critical graph of odd order with minimum degree at least 2.

scholarly journals A Better Lower Bound on Average Degree of 4-List-Critical Graphs

10.37236/5971 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Lower Bound ◽  
Short Note ◽  
Average Degree ◽  
Critical Graphs ◽  
Critical Graph

This short note proves that every non-complete $k$-list-critical graph has average degree at least $k-1 + \frac{k-3}{k^2-2k+2}$. This improves the best known bound for $k = 4,5,6$. The same bound holds for online $k$-list-critical graphs.

scholarly journals Structural Properties of Connected Domination Critical Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2568
Author(s):  
Norah Almalki ◽  
Pawaton Kaemawichanurat
Keyword(s):  
Structural Properties ◽  
Perfect Matching ◽  
Minimum Degree ◽  
Domination Number ◽  
Free Graph ◽  
Critical Graphs ◽  
Critical Graph ◽  
Odd Order ◽  
Connected Domination Number ◽  
Even Order

A graph G is said to be k-γc-critical if the connected domination number γc(G) is equal to k and γc(G+uv)<k for any pair of non-adjacent vertices u and v of G. Let ζ be the number of cut vertices of G and let ζ0 be the maximum number of cut vertices that can be contained in one block. For an integer ℓ≥0, a graph G is ℓ-factor critical if G−S has a perfect matching for any subset S of vertices of size ℓ. It was proved by Ananchuen in 2007 for k=3, Kaemawichanurat and Ananchuen in 2010 for k=4 and by Kaemawichanurat and Ananchuen in 2020 for k≥5 that every k-γc-critical graph has at most k−2 cut vertices and the graphs with maximum number of cut vertices were characterized. In 2020, Kaemawichanurat and Ananchuen proved further that, for k≥4, every k-γc-critical graphs satisfies the inequality ζ0(G)≤mink+23,ζ. In this paper, we characterize all k-γc-critical graphs having k−3 cut vertices. Further, we establish realizability that, for given k≥4, 2≤ζ≤k−2 and 2≤ζ0≤mink+23,ζ, there exists a k-γc-critical graph with ζ cut vertices having a block which contains ζ0 cut vertices. Finally, we proved that every k-γc-critical graph of odd order with minimum degree two is 1-factor critical if and only if 1≤k≤2. Further, we proved that every k-γc-critical K1,3-free graph of even order with minimum degree three is 2-factor critical if and only if 1≤k≤2.

A maximum degree theorem for diameter-2-critical graphs

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Teresa Haynes ◽  
Michael Henning ◽  
Lucas Merwe ◽  
Anders Yeo
Keyword(s):  
Graph Theory ◽  
Maximum Degree ◽  
Discrete Math ◽  
Bipartite Graphs ◽  
Extremal Graphs ◽  
Critical Graphs ◽  
Critical Graph ◽  
Complete Bipartite Graphs

AbstractA graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ⌊n 2/4⌋ and that the extremal graphs are the complete bipartite graphs K ⌊n/2⌋,⌊n/2⌉. Fan [Discrete Math. 67 (1987), 235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98] proved the conjecture for n > n 0 where n 0 is a tower of 2’s of height about 1014. The conjecture has yet to be proven for other values of n. Let Δ denote the maximum degree of G. We prove the following maximum degree theorems for diameter-2-critical graphs. If Δ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and Δ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.

Bicritical domination and double coalescence of graphs

2016 ◽  
Vol 23 (3) ◽  
pp. 399-404 ◽  
Author(s):  
Marcin Krzywkowski ◽  
Doost Ali Mojdeh
Keyword(s):  
Domination Number ◽  
Edge Connectivity ◽  
Critical Graphs

AbstractA graph is bicritical if the removal of any pair of vertices decreases the domination number. We study the properties of bicritical graphs and their relation with critical graphs, and we obtain results for bicritical graphs with edge connectivity two or three. We also generalize the notion of the coalescence of two graphs and investigate the bicriticality of such graphs.

scholarly journals NEIGHBOURHOODS OF INDEPENDENT SETS FOR (a,b,k)-CRITICAL GRAPHS

2008 ◽  
Vol 77 (2) ◽  
pp. 277-283 ◽  
Author(s):  
SIZHONG ZHOU ◽  
YANG XU
Keyword(s):  
Independent Sets ◽  
Sufficient Condition ◽  
Critical Graphs ◽  
Critical Graph

AbstractLet G be a graph of order n. Let a, b and k be nonnegative integers such that 1≤a≤b. A graph G is called an (a,b,k)-critical graph if after deleting any k vertices of G the remaining graph of G has an [a,b]-factor. We provide a sufficient condition for a graph to be (a,b,k)-critical that extends a well-known sufficient condition for the existence of a k-factor.

scholarly journals Fractional Matchings, Component-Factors and Edge-Chromatic Critical Graphs

2021 ◽  
Author(s):  
Antje Klopp ◽  
Eckhard Steffen
Keyword(s):  
Upper Bounds ◽  
Matching Number ◽  
Critical Graphs ◽  
Critical Graph ◽  
Minimum Number

AbstractThe first part of the paper studies star-cycle factors of graphs. It characterizes star-cycle factors of a graph G and proves upper bounds for the minimum number of $$K_{1,2}$$ K 1 , 2 -components in a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor of a graph G. Furthermore, it shows where these components are located with respect to the Gallai–Edmonds decomposition of G and it characterizes the edges which are not contained in any $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor of G. The second part of the paper proves that every edge-chromatic critical graph G has a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor, and the number of $$K_{1,2}$$ K 1 , 2 -components is bounded in terms of its fractional matching number. Furthermore, it shows that for every edge e of G, there is a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor F with $$e \in E(F)$$ e ∈ E ( F ) . Consequences of these results for Vizing’s critical graph conjectures are discussed.

Sign in / Sign up

Export Citation Format

Share Document